108 ideas
1502 | Parmenides was much more cautious about accepting ideas than his predecessors [Simplicius on Parmenides] |
6052 | Definitions identify two concepts, so they presuppose identity [McGinn] |
9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
6064 | Regresses are only vicious in the context of an explanation [McGinn] |
6088 | Truth is a method of deducing facts from propositions [McGinn] |
6084 | 'Snow does not fall' corresponds to snow does fall [McGinn] |
6085 | The idea of truth is built into the idea of correspondence [McGinn] |
6083 | The coherence theory of truth implies idealism, because facts are just coherent beliefs [McGinn] |
6086 | Truth is the property of propositions that makes it possible to deduce facts [McGinn] |
6087 | Without the disquotation device for truth, you could never form beliefs from others' testimony [McGinn] |
10098 | The 'power set' of A is all the subsets of A [George/Velleman] |
10099 | The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}} [George/Velleman] |
10101 | Cartesian Product A x B: the set of all ordered pairs in which a∈A and b∈B [George/Velleman] |
10103 | Grouping by property is common in mathematics, usually using equivalence [George/Velleman] |
10104 | 'Equivalence' is a reflexive, symmetric and transitive relation; 'same first letter' partitions English words [George/Velleman] |
10096 | Even the elements of sets in ZFC are sets, resting on the pure empty set [George/Velleman] |
10097 | Axiom of Extensionality: for all sets x and y, if x and y have the same elements then x = y [George/Velleman] |
10100 | Axiom of Pairing: for all sets x and y, there is a set z containing just x and y [George/Velleman] |
17900 | The Axiom of Reducibility made impredicative definitions possible [George/Velleman] |
10109 | ZFC can prove that there is no set corresponding to the concept 'set' [George/Velleman] |
10108 | As a reduction of arithmetic, set theory is not fully general, and so not logical [George/Velleman] |
10111 | Asserting Excluded Middle is a hallmark of realism about the natural world [George/Velleman] |
6051 | In 'x is F and x is G' we must assume the identity of x in the two statements [McGinn] |
6055 | Both non-contradiction and excluded middle need identity in their formulation [McGinn] |
6059 | Identity is unitary, indefinable, fundamental and a genuine relation [McGinn] |
6042 | The quantifier is overrated as an analytical tool [McGinn] |
6067 | Existential quantifiers just express the quantity of things, leaving existence to the predicate 'exists' [McGinn] |
6069 | 'Partial quantifier' would be a better name than 'existential quantifier', as no existence would be implied [McGinn] |
6068 | We need an Intentional Quantifier ("some of the things we talk about.."), so existence goes into the proposition [McGinn] |
10129 | A 'model' is a meaning-assignment which makes all the axioms true [George/Velleman] |
10105 | Differences between isomorphic structures seem unimportant [George/Velleman] |
10119 | Consistency is a purely syntactic property, unlike the semantic property of soundness [George/Velleman] |
10126 | A 'consistent' theory cannot contain both a sentence and its negation [George/Velleman] |
10120 | Soundness is a semantic property, unlike the purely syntactic property of consistency [George/Velleman] |
10127 | A 'complete' theory contains either any sentence or its negation [George/Velleman] |
10106 | Rational numbers give answers to division problems with integers [George/Velleman] |
10102 | The integers are answers to subtraction problems involving natural numbers [George/Velleman] |
10107 | Real numbers provide answers to square root problems [George/Velleman] |
9946 | Logicists say mathematics is applicable because it is totally general [George/Velleman] |
10125 | The classical mathematician believes the real numbers form an actual set [George/Velleman] |
17899 | Second-order induction is stronger as it covers all concepts, not just first-order definable ones [George/Velleman] |
10128 | The Incompleteness proofs use arithmetic to talk about formal arithmetic [George/Velleman] |
17902 | A successor is the union of a set with its singleton [George/Velleman] |
10133 | Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle [George/Velleman] |
10130 | Set theory can prove the Peano Postulates [George/Velleman] |
10089 | Talk of 'abstract entities' is more a label for the problem than a solution to it [George/Velleman] |
10131 | If mathematics is not about particulars, observing particulars must be irrelevant [George/Velleman] |
10092 | In the unramified theory of types, the types are objects, then sets of objects, sets of sets etc. [George/Velleman] |
10094 | The theory of types seems to rule out harmless sets as well as paradoxical ones. [George/Velleman] |
10095 | Type theory has only finitely many items at each level, which is a problem for mathematics [George/Velleman] |
17901 | Type theory prohibits (oddly) a set containing an individual and a set of individuals [George/Velleman] |
10114 | Bounded quantification is originally finitary, as conjunctions and disjunctions [George/Velleman] |
10134 | Much infinite mathematics can still be justified finitely [George/Velleman] |
10123 | The intuitionists are the idealists of mathematics [George/Velleman] |
10124 | Gödel's First Theorem suggests there are truths which are independent of proof [George/Velleman] |
6070 | Existence is a primary quality, non-existence a secondary quality [McGinn] |
448 | No necessity could produce Being either later or earlier, so it must exist absolutely or not at all [Parmenides] |
447 | Being must be eternal and uncreated, and hence it is timeless [Parmenides] |
449 | Being is not divisible, since it is all alike [Parmenides] |
1503 | There is no such thing as nothing [Parmenides] |
445 | The realm of necessary non-existence cannot be explored, because it is unknowable [Parmenides] |
21820 | Parmenides at least saw Being as the same as Nous, and separate from the sensed realm [Parmenides, by Plotinus] |
6062 | Existence can't be analysed as instantiating a property, as instantiation requires existence [McGinn] |
6065 | We can't analyse the sentence 'something exists' in terms of instantiated properties [McGinn] |
452 | All our concepts of change and permanence are just names, not the truth [Parmenides] |
6082 | If causal power is the test for reality, that will exclude necessities and possibilities [McGinn] |
6075 | Facts are object-plus-extension, or property-plus-set-of-properties, or object-plus-property [McGinn] |
1504 | Something must be unchanging to make recognition and knowledge possible [Aristotle on Parmenides] |
6058 | Identity propositions are not always tautological, and have a key epistemic role [McGinn] |
6053 | Identity is as basic as any concept could ever be [McGinn] |
6043 | Type-identity is close similarity in qualities [McGinn] |
6044 | Qualitative identity is really numerical identity of properties [McGinn] |
6046 | Qualitative identity can be analysed into numerical identity of the type involved [McGinn] |
6045 | It is best to drop types of identity, and speak of 'identity' or 'resemblance' [McGinn] |
6066 | Existence is a property of all objects, but less universal than self-identity, which covers even conceivable objects [McGinn] |
6054 | Sherlock Holmes does not exist, but he is self-identical [McGinn] |
6047 | All identity is necessary, though identity statements can be contingently true [McGinn] |
6049 | Leibniz's Law says 'x = y iff for all P, Px iff Py' [McGinn] |
6048 | Leibniz's Law is so fundamental that it almost defines the concept of identity [McGinn] |
6050 | Leibniz's Law presupposes the notion of property identity [McGinn] |
444 | The first way of enquiry involves necessary existence [Parmenides] |
450 | Necessity sets limits on being, in order to give it identity [Parmenides] |
6080 | Modality is not objects or properties, but the type of binding of objects to properties [McGinn] |
6079 | If 'possible' is explained as quantification across worlds, there must be possible worlds [McGinn] |
451 | Thinking implies existence, because thinking depends on it [Parmenides] |
1506 | Parmenides treats perception and intellectual activity as the same [Theophrastus on Parmenides] |
3058 | Only reason can prove the truth of facts [Parmenides] |
6081 | Necessity and possibility are big threats to the empiricist view of knowledge [McGinn] |
6071 | Scepticism about reality is possible because existence isn't part of appearances [McGinn] |
10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
6077 | Semantics should not be based on set-membership, but on instantiation of properties in objects [McGinn] |
6074 | Clearly predicates have extensions (applicable objects), but are the extensions part of their meaning? [McGinn] |
555 | People who say that the cosmos is one forget that they must explain movement [Aristotle on Parmenides] |
5081 | There could be movement within one thing, as there is within water [Aristotle on Parmenides] |
1509 | The one can't be divisible, because if it was it could be infinitely divided down to nothing [Parmenides, by Simplicius] |
20900 | Defenders of the One say motion needs the void - but that is not part of Being [Parmenides, by Aristotle] |
226 | The one is without any kind of motion [Parmenides] |
1505 | Reason sees reality as one, the senses see it as many [Aristotle on Parmenides] |
453 | Reality is symmetrical and balanced, like a sphere, with no reason to be greater one way rather than another [Parmenides] |
1792 | He taught that there are two elements, fire the maker, and earth the matter [Parmenides, by Diog. Laertius] |
5115 | It is feeble-minded to look for explanations of everything being at rest [Aristotle on Parmenides] |
13217 | The void can't exist, and without the void there can't be movement or separation [Parmenides, by Aristotle] |
22918 | What could have triggered the beginning [of time and being]? [Parmenides] |
1791 | He was the first person to say the earth is spherical [Parmenides, by Diog. Laertius] |
1794 | He was the first to discover the identity of the Morning and Evening Stars [Parmenides, by Diog. Laertius] |
6072 | If Satan is the most imperfect conceivable being, he must have non-existence [McGinn] |
6073 | I think the fault of the Ontological Argument is taking the original idea to be well-defined [McGinn] |